Why does the right leg's bone in my rig move when I move the left?
Most likely, this is a Weight Painting issue. 
You may have some spill-over weight that reaches to the other side of your model. 



To check for this, go into Weight Paint Mode and go through the Vertex Groups on one side, one by one. Check for weights on the other side of the symmetry line. You can make this visually easier, by choosing Show Active Weights > Active. With this, vertices that are not in the selected Vertex Group are shown in black. 

I would suggest Normalizing your weight beforehand so that the colors you see correspond to the actual weights on the vertices.

If you don't know exactly why there would be a difference, watch this rigging tutorial. He goes into detail on the hows and whys: DanPro Skinning Basics on Youtube The series is very good.

After taking a look at your file, I discovered that your bone-naming is incorrect. However you mirrored your weights over, you ended up with one empty set (no weights at all) and one that has the weighting of both sides. 

A tip for the future: if you use a Mirror-Modifier on your mesh while building it, you can leave that on through the weight painting phase and it will transfer your weight groups over automatically. That saves a lot of work. The only caveat is that vertex paint will only work on the side that has the actual geometry, not the one with the virtual vertices.



If you take a look at your bones or vertex groups, there is a series of .R leg bones. These should, of course, have their corresponding .L bones. 

To fix this, I would suggest removing one half of the rig and mirroring the remaining half over a second time. 

If your character was aligned head to tail with the Y-Axis, you could use the symmetrize function, since it is not, you need to do a bit more manual work.

To achieve this, you can do the following:


Go through your right side Vertex Groups and make sure the weight in them is only influencing the right side. Remove all other weight. If you have Show Active Weight on (see above) then the rest of the mesh should be black.
Delete the bones with the number suffix.
Put the 3D-Cursor on the X-Axis (Shift + S > Snap Cursor to Center)
Make sure your transformation origin is at the 3D-Cursor-
Select the bones you need to mirror, for the dinosaur those would be the arms and legs.
Hit Ctrl + M followed by Y to mirror along the Y-Axis
Correct your bone naming from .R to .L
With your bones in place, all that is left is to mirror the existing Vertex Groups over to the other side. There is a detailed guide to do this.


Here is a link to your file, I mirrored your weights over